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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,50,3]))
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pari:[g,chi] = znchar(Mod(77,3600))
| Modulus: | \(3600\) | |
| Conductor: | \(3600\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(60\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{3600}(77,\cdot)\)
\(\chi_{3600}(317,\cdot)\)
\(\chi_{3600}(533,\cdot)\)
\(\chi_{3600}(797,\cdot)\)
\(\chi_{3600}(1013,\cdot)\)
\(\chi_{3600}(1037,\cdot)\)
\(\chi_{3600}(1253,\cdot)\)
\(\chi_{3600}(1517,\cdot)\)
\(\chi_{3600}(1733,\cdot)\)
\(\chi_{3600}(1973,\cdot)\)
\(\chi_{3600}(2237,\cdot)\)
\(\chi_{3600}(2453,\cdot)\)
\(\chi_{3600}(2477,\cdot)\)
\(\chi_{3600}(3173,\cdot)\)
\(\chi_{3600}(3197,\cdot)\)
\(\chi_{3600}(3413,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((3151,901,2801,577)\) → \((1,-i,e\left(\frac{5}{6}\right),e\left(\frac{1}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3600 }(77, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) |
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sage:chi.jacobi_sum(n)
